Fill in the blanks in the proof of the following theorem.
THEOREM: (A ∩ B) × C = (A × C) ∩ (B × C)
Proof: Let (x,y) ∈ (A ∩ B) × C. Then x ∈ ___ and y ∈ ___ . Since x ∈ A ∩ B, x ∈ ___ and x ∈ ___ . Thus (x,y) ∈ ___ and (x,y) ∈ . Hence (x,y) ∈ (A × C) ∩ (B × C), so ______ ⊆ ___ On the other hand, suppose that (x,y) ∈ . Then (x,y) ∈ and (x,y) ∈ . Since (x,y) ∈A × C , x ∈ ___ and y ∈ Since (x,y) ∈ B × C, ____ and ___ . Thus x ∈ A ∩ B , so ___ ∈ _____ and _______
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